This paper proposes a natural language translation method for machine-verifiable formal proofs that leverages the informalization (verbalization of formal language proof steps) and summarization capabilities of LLMs. For evaluation, it was applied to formal proof data created in accordance with natural language proofs taken from an undergraduate-level textbook, and the quality of the generated natural language proofs was analyzed in comparison with the original natural language proofs. Furthermore, we will demonstrate that this method can output highly readable and accurate natural language proofs by applying it to existing formal proof library of the Lean proof assistant.

Abstract We systematically study conservation theorems on theories of semi-classical arithmetic, which lie in-between classical arithmetic $\mathsf {PA}$ and intuitionistic arithmetic $\mathsf {HA}$ . Using a generalized negative translation, we first provide a structured proof of the fact that $\mathsf {PA}$ is $\Pi _{k+2}$ -conservative over $\mathsf {HA} + {\Sigma _k}\text {-}\mathrm {LEM}$ where ${\Sigma _k}\text {-}\mathrm {LEM}$ is the axiom scheme of the law-of-excluded-middle restricted to formulas in $\Sigma _k$ . In addition, we show that this conservation theorem is optimal in the sense that for any semi-classical arithmetic T, if $\mathsf {PA}$ is $\Pi _{k+2}$ -conservative over T, then ${T}$ proves ${\Sigma _k}\text {-}\mathrm {LEM}$ . In the same manner, we also characterize conservation theorems for other well-studied classes of formulas by fragments of classical axioms or rules. This reveals the entire structure of conservation theorems with respect to the arithmetical hierarchy of classical principles.

Abstract Akama et al. [1] systematically studied an arithmetical hierarchy of the law of excluded middle and related principles in the context of first-order arithmetic. In that paper, they first provide a prenex normal form theorem as a justification of their semi-classical principles restricted to prenex formulas. However, there are some errors in their proof. In this paper, we provide a simple counterexample of their prenex normal form theorem [1, Theorem 2.7], then modify it in an appropriate way which still serves to largely justify the arithmetical hierarchy. In addition, we characterize a variety of prenex normal form theorems by logical principles in the arithmetical hierarchy. The characterization results reveal that our prenex normal form theorems are optimal. For the characterization results, we establish a new conservation theorem on semi-classical arithmetic. The theorem generalizes a well-known fact that classical arithmetic is $\Pi _2$ -conservative over intuitionistic arithmetic.

Abstract We investigate two weak fragments of the double negation shift schema, which are motivated, respectively, from Spector’s consistency proof of ACA0 and from the negative translation of RCA0, as well as double negated variants of logical principles. Their interrelations over both intuitionistic arithmetic and analysis are completely solved.

Recently many studies on ubiquitous computing have been done. In the studies, ubiquitous ID architecture is one of the most important ones. In this study, we propose a new method for connecting to objects with RFTDs through the Internet. At first, we introduce a simple application called "Portable Environment Communicator". We try to apply the proposed method to it. Second, we propose a new name resolver architecture for portable objects, and a new method for selecting the most appropriate wireless base station around the objects, which is connected to the Internet. In the paper, the selection method is carried out using the supper arrangement method, distance field space model, and fuzzy measure theory.